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Encyclopedia > Eisenstein series

In mathematics, Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ... Modular form - Wikipedia /**/ @import /skins-1. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ...

Contents

Eisenstein series for the modular group

The real part of G_6 as a function of q on the unit disk.
The real part of G_6 as a function of q on the unit disk.
The imaginary part of G_6 as a function of q on the unit disk.
The imaginary part of G_6 as a function of q on the unit disk.

Let τ be a complex number with strictly positive imaginary part. Define the holomorphic Eisenstein series G2k(τ) of weight 2k, where kgeq 2 is an integer, by the following series: Weierstrass elliptic functions invarient g3, real part (600x600 pixels) Detailed description This image shows the real part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... Weierstrass elliptic functions invarient g3, real part (600x600 pixels) Detailed description This image shows the real part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... Weierstrass elliptic functions invarient g3, imaginary part (600x600 pixels) Detailed description This image shows the imaginary part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... Weierstrass elliptic functions invarient g3, imaginary part (600x600 pixels) Detailed description This image shows the imaginary part of the Weierstrass elliptic functions invarient g3=140 G6 as a function of the square of the nome on the unit disk |q| &lt 1. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...

G_{2k}(tau) = sum_{ (m,n) neq (0,0)} frac{1}{(m+ntau )^{2k}}.

This series absolutely converges to a holomorphic function of τ in the upper half-plane and its Fourier expansion given below shows that it extends to a holomorhic function at tau=iinfty. It is a remarkable fact that the Eisenstein series is a modular form. Indeed, the key property is its SL_2(mathbb{Z})-invariance. Explicitly if a,b,c,d in mathbb{Z} and ad − bc = 1 then In mathematics, a series is a sum of a sequence of terms. ... this article is useless. ... Modular form - Wikipedia /**/ @import /skins-1. ...

G_{2k} left( frac{ atau +b}{ ctau + d} right) = (ctau +d)^{2k} G_{2k}(tau)

and G2k is therefore a modular form of weight 2k. Note that it is important to assume that kgeq 2, otherwise it would be illegitimate to change the order of summation, and the SL_2(mathbb{Z})-invariance would not hold. In fact, there are no nontrivial modular forms of weight 2. Nevertheless, an analogue of the holomorphic Eisenstein series can be defined even for k = 1, although it would only be a near modular form.


Relation to modular invariants

The modular invariants g2 and g3 of an elliptic curve are given by the first two terms of the Eisenstein series as In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ...

g2 = 60G4

and

g3 = 140G6

The article on modular invariants provides expressions for these two functions in terms of theta functions. In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... In mathematics, theta functions are special functions of several complex variables. ...


Recurrence relation

Any holomorphic modular form for the modular group can be written as a polynomial in G4 and G6. Specifically, the higher order G2k's can be written in terms of G4 and G6 through a recurrence relation. Let dk = (2k + 3)k!G2k + 4. Then the dk satisfy the relation

sum_{k=0}^n {n choose k} d_k d_{n-k} = frac{2n+9}{3n+6}d_{n+2}

for all nge 0. Here, {n choose k} is the binomial coefficient and d0 = 3G4 and d1 = 5G6. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ...


The dk occur in the series expansion for the Weierstrass's elliptic functions: In mathematics, Weierstrasss elliptic functions are a standard type of elliptic functions (the other is the Jacobis elliptic functions). ...

wp(z) =frac{1}{z^2} + z^2 sum_{k=0}^infty frac {d_k z^{2k}}{k!} =frac{1}{z^2} + sum_{k=1}^infty (2k+1) G_{2k+2} z^{2k}

Fourier series

Define q = e2πiτ. (Some older books define q to be the nome q = eiπτ, but q = e2πiτ is now standard in number theory.) Then the Fourier series of the Eisenstein series is In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

G_{2k}(tau) = 2zeta(2k) left(1+c_{2k}sum_{n=1}^{infty} sigma_{2k-1}(n)q^{n} right)

where the Fourier coefficients c2k are given by In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function (often taken to have period 2π — in a sense, the simplest case) as a sum of periodic functions of the form which are harmonics of ei x. ...

c_{2k} = frac{(2pi i)^{2k}}{(2k-1)! zeta(2k)} = frac {-4k}{B_{2k}} .

Here, Bn are the Bernoulli numbers, ζ(z) is Riemann's zeta function and the sigma function σp(n) is the sum of the pth powers of the divisors of n. In particular, one has In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. ... In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ... See divisor function for arithmetic functions σa(n), busy beaver for Rados sigma function. ...

G_4(tau)=frac{pi^4}{45} left[ 1+ 240sum_{n=1}^infty sigma_3(n) q^{n} right]

and

G_6(tau)=frac{2pi^6}{945} left[ 1- 504sum_{n=1}^infty sigma_5(n) q^{n} right]

Note the summation over q can be resummed as a Lambert series; that is, one has In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resummed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of with the constant function : This series may be inverted by means...

sum_{n=1}^{infty} q^n sigma_a(n) = sum_{n=1}^{infty} frac{n^a q^n}{1-q^n}

for arbitrary complex |q| ≤ 1 and a. When working with the q-expansion of the Eisenstein series, the alternate notation In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

is frequently introduced.


Identities involving Eisenstein series

Products of Eisenstein series

Eisenstein series form the most explicit examples of modular forms for the full modular group Since the space of modular forms of weight 2k has dimension 1 for 2k = 4,6,8,10,14 different products of Eisenstein series having those weights have to be proportional. Thus we obtain the identities: Modular form - Wikipedia /**/ @import /skins-1. ...

Using the q-expansions of the Eisenstein series given above, they may be restated as identities involving the sums of powers of divisors:

hence

and similarily for the others. Perhaps, even more interestingly, the theta function of an eight-dimensional even unimodular lattice Γ is a modular form of weight 4 for the full modular group, which gives the following identities: In mathematics, theta functions are special functions of several complex variables. ...

for the number rΓ(n) of vectors of the squared length 2n in the root lattice of the type E8. In mathematics, the E8 lattice is a special lattice in R8. ...


Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer n as a sum of two, four, and eight squares in terms of the divisors of n. In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...


Ramanujan identities

Ramanujan gave several interesting identities between the first few Eisenstein series involving differentiation. Let Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...

and

and

then

and

and

These identities yield correspondent arithmetical convolution identities involving the sum-of-divisor function, as for example In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...

Other identities of this type, but not directly related to the preceding relations between L, M and N functions, have been proved by Ramanujan and Melfi, as for example Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ... Giuseppe Melfi (June 11, 1967), Italian mathematician. ...

For a comprehensive list of convolution identities involving sum-of-divisors functions and related topics see

  • S. Ramanujan, On certain arithmetical functions, pp 136-162, reprinted in Collected Papers, (1962), Chelsea, New York.
  • Heng Huat Chan and Yau Lin Ong, On Eisenstein Series, (1999) Proceedings of the Amer. Math. Soc. 127(6) pp.1735-1744
  • G. Melfi, On some modular identities, in Number Theory, Diophantine, Computational and Algebraic Aspects: Proceedings of the International Conference held in Eger, Hungary. Walter de Grutyer and Co. (1998), 371-382.

Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ... Giuseppe Melfi (June 11, 1967), Italian mathematician. ...

Generalizations

Automorphic forms generalize the idea of modular forms for general Lie groups; and Eisenstein series generalize in a similar fashion. In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...


Defining OK to be the ring of integers of a totally real algebraic number field K, one then defines the Hilbert-Blumenthal modular group as PSL(2,OK). One can then associate an Eisenstein series to every cusp of the Hilbert-Blumenthal modular group. In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn−1 + ··· + an −1x + an = 0 where n is a positive integer called... In number theory, a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers. ... In mathematics, a Hilbert modular form is a generalization of the elliptic modular forms, to functions of two or more variables. ... In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion of the constant coefficient a0. ...


References

  • Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
  • Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0
  • Henryk Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics), America Mathematical Society, Providence, RI ISBN 0-8218-3160-7 (See chapter 3)
  • Serre, Jean-Pierre, A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.

  Results from FactBites:
 
Vignettes on automorphic and modular forms, representations, L-functions, and number theory (1519 words)
A version of it will appear in the AIM conference volume on Eisenstein series, likely as a too-long appendix to a paper of Michael Harris', namely "A simple proof of rationality of Siegel-Weil Eisenstein series" [http://www.math.jussieu.fr/~harris/SW.pdf] which does explain something of the context.
The absolutely simplest case: spherical Eisenstein series for SL(2,Z), of course, assuming basic results from the theory of the constant term, paying attention to the proper notion of truncation.
Used in reduction of the non-maximal parabolic case to the maximal parabolic case in treatment of Eisenstein series, and in the proof (for the Selberg-Bernstein argument for meromorphic continuation) that a composition of weakly holomorphic morphisms of topological vectorspaces maps is again weakly holomorphic.
  More results at FactBites »


 

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